SOLUTION: A nationwide sample of influential Republicans and Democrats was asked as a part of a comprehensive survey whether they favored lowering environmental standards so that high-sulfur

Question 1150934: A nationwide sample of influential Republicans and Democrats was asked as a part of a comprehensive survey whether they favored lowering environmental standards so that high-sulfur coal could be burned in coal-fired power plants. The results were:
Number sampled : 1,000 (republican) , 800 (democrats)
Number in favor: 200 (republican) , 168 (democrats)
At the .02 level of significance, can we conclude that there is a larger proportion of Democrats in favor of lowering the standards?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Let group 1 be the sample of Republicans. Group 2 will be the sample of Democrats.

H0:

(equivalent to

) is the null hypothesis
H1:

(equivalent to

) is the alternative hypothesis
where,

= population proportion of Republicans in favor of lowering the standards

= population proportion of Democrats in favor of lowering the standards

We are conducting a left tailed test because of the "less than" sign in the alternative hypothesis.

We are given

as the two counts of successes (ie the number of people who responded in favor for each group). And also, we have these sample sizes

Which helps us find the sample proportions for each group

is the sample proportion of Republicans in favor of lowering the standards

is the sample proportion of Democrats in favor of lowering the standards
Based on these two p-hat values, it does look like there is a higher proportion of Democrats in favor of lowering the standards. However, let's continue with the hypothesis test.

Compute the pooled proportion. The bar over the 'p' indicates an average.

Now onto the Standard Error.

is the approximate standard error

Compute the test statistic

Use a Z table similar to this one
http://www.z-table.com/
To find that

The P-value is approximately 0.3015

This is much larger than alpha = 0.02, so we fail to reject the null hypothesis.
We do not have enough statistically significant evidence to overturn or reject the null.
Therefore, we must accept that the population proportions p1 and p2 are the same.
In other words, the proportion of people in favor of lowering the standards are the same for each group.

Going back to the question "can we conclude that there is a larger proportion of Democrats in favor of lowering the standards?", the answer is "no, we cannot conclude there is a larger proportion of Democrats in favor of lowering the standards".

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